High-resolution Interferometric Diagnostics for Ultrashort Pulses

8. QUANTUM-PATH INTERFEROMETRY IN HIGH-HARMONIC GENERATIONture which could be systematically applied to the entire dataset. To do so, I included a quadraticterm in the power series expansion of the control-field dependent action:S (j ) (ω) ≈ S (j )D + ¯θ T ¯S (j )C + 1 2 ¯θ T (j ) ¯θ . (8.17)Here (j ) is a matrix which describes all combinations of quadratic self- and cross- couplingsbetween the control fields. I then obtained (j ) by fitting (8.17) to the exact quantum orbit analysis.In general, ¯θ T (j ) ¯θ describes a paraboloid, the contours of which are arbitrarily aligned tothe coordinate axes. The direction and magnitudes of the principal axes are given by diagonalization.Here, I not concerned with the details of the geometry, but only the magnitude of thecurvature. So, without loss of generality one may choose a co-ordinate system in which (j ) isdiagonal with largest eigenvalue ( (j ) ) 11 . This eigenvalue is equal to the largest quadratic coefficientover all possible directions in ¯θ space, and therefore forms a convenient measure of thecurvature of the action. This quantity is plotted for the long and short first-order trajectories inFig. 8.7, and for the long trajectory (the worst case) is around 1000. One obtains a feeling for thesignificance of this number by noting that the erroneous phase introduced by the nonlinearity isequal to 1/2( (j ) ) 11 θ1 2 . The control field amplitude at which the error equals one radian is therefore2/( (j ) ) 11 ≈ 0.045 for the monochromatic example used throughout thischapter.8.5.2 Choice of control-field amplitude scan rangeThe previous section showed that, away from cutoff, nonlinearity of action change is the dominantdistortion. The other factor that influences the width of the orbits in control-field-sensitivityspace is the intrinsic resolution in that space, determined by the range of control fields over whichD(ω; ¯θ ) is known. One may represent this mathematically via a window function ( ¯θ ) which isunity at the origin ¯θ = 0 and smoothly decays when approaching the edge of the sampled rangeof control field amplitudes. For the discussion which follows it is necessary to choose a particularform for the window function — here I use an isotropic Gaussian function ( ¯θ )=exp(− | ¯θ | 22w 2 ) (8.18)202